Announcements Topics: -sections (differentiation rules), 5.6, and 5.7 * Read these sections and study solved examples in your textbook! Work On: -Practice problems from the textbook and assignments from the coursepack as assigned on the course web page (under the link “SCHEDULE + HOMEWORK”)

Chain Rule “derivative of the outer function evaluated at the inner function times the derivative of the inner function” Example: Differentiate the following. (a) (b)

Chain Rule Example: Using implicit differentiation, determine for

Chain Rule Example: The number of mosquitoes (M) that end up in a room is a function of how far the window is open (W, in square centimetres) according to The number of bites (B) depends on the number of mosquitoes according to Find the derivative of B as a function of W.

Derivative of the Natural Exponential Function Definition: The number e is the number for which Natural Exponential Function:

Derivative of the Natural Exponential Function Note: This definition states that the slope of the tangent to the curve at (0,1) is exactly 1, i.e.

Derivative of the Natural Exponential Function If then Proof: Note: The slope of a tangent line to the curve is equal to the value of the function at that point. Note: The slope of a tangent line to the curve is equal to the value of the function at that point.

Derivatives of Exponential Functions If then Example: Differentiate. (a) (b) (c)

Derivatives of Logarithmic Functions If then Example 1: Differentiate. (a)(b) Example 2: Determine the equation of the tangent line to the curve at the point

Derivatives of Trigonometric Functions

Example: Find the derivative of each. (a)(b) (c)

Derivatives of Inverse Trig Functions Example 1: Differentiate. (a) Example 2: Prove

The Second Derivative The derivative of the derivative is called the second derivative. the second derivative of f =

The Second Derivative f” provides information about f’ and f: When f’’ is positive, f’ is increasing, i.e., the rate at which f is changing is increasing. When f’’ is positive, the slopes of the tangents to the graph of f are increasing and the graph of f is concave up.

The Second Derivative f” provides information about f’ and f: When f’’ is negative, f’ is decreasing, i.e., the rate at which f is changing is decreasing. When f’’ is negative, the slopes of the tangents to the graph of f are decreasing and the graph of f is concave down.

The Second Derivative When the graph of f changes concavity at a point in the domain of f, this point is called an inflection point. Note: At an inflection point, f”=0 or f’’ D.N.E.

The Second Derivative Example: Gamma Distribution Find the first and second derivatives of g(x) and use them to sketch a graph.

Approximating Functions with Polynomials Polynomials have many nice properties and are generally very easy to work with. For this reason, it is often useful to approximate more complicated functions with polynomials in order to simplify calculations. Linear functions are the simplest polynomials and can be used to represent a more complicated function in many situations.

Linear Approximations The secant line connecting points (a, f(a)) and (b, f(b)) on the graph of f(x) provides a decent linear approximation to f(x) for x-values in the interval [a, b]. The tangent line to the graph of f(x) at (a, f(a)) provides the best linear (straight line) approximation to f(x) near x=a. Linear (or tangent line) approximation to f(x) around x=a:

Linear Approximations Example: Let Find the secant line approximation to f(x) for x-values between 1 and 4. Use this to approximate both and. Compare to the actual values. Using Technology:

Linear Approximations Example: Let Find the tangent line approximation to f(x) at x=1. Use this to approximate both and. Compare to the actual values. Using Technology:

Quadratic Approximations The tangent line (linear) approximation matches the value and the slope of the function at x=1. The quadratic approximation matches the value, the slope, and the curvature (concavity) of the function at x=1. We can obtain a more accurate approximation by using polynomials of higher degrees.

Quadratic Approximations To find the quadratic approximation to a function at the base point a, we match the value, the first derivative, and the second derivative at the point a. Quadratic approximation to around x=a: because the degree is 2

Quadratic Approximations Example: Let Find the quadratic approximation to f(x) around x=1.

The Taylor Polynomial Suppose the first n derivatives of the function f are defined at x=a. Then the Taylor polynomial of degree n matching the values of the first n derivatives is n th derivative

The Taylor Polynomial Example: Find the 1 st, 3 rd, and 5 th degree Taylor polynomials for near x=0.

The Taylor Polynomial